Approximating Minimum Reset Sequences

We consider the problem of finding minimum reset sequences in synchronizing automata. The well-known Cerný conjecture states that every n-state synchronizing automaton has a reset sequence with length at most (n - 1)2. While this conjecture gives an upper bound on the length of every reset sequence, it does not directly address the problem of finding the shortest reset sequence. We call this the MINIMUM RESET SEQUENCE (MRS) problem. We give an O(kmnk + n4/k)-time ⌈n-1/k-1⌉-approximation for the MRS problem for any k ≥ 2. We also show that our analysis is tight. When k = 2 our algorithm reduces to Eppstein's algorithm and yields an (n-1)-approximation. When k = n our algorithm is the familiar exponential-time, exact algorithm. We define a nontrivial class of MRS which we call STACK COVER. We show that STACK COVER naturally generalizes two classic optimization problems: MIN SET COVER and SHORTEST COMMON SUPERSEQUENCE. Both these problems are known to be hard to approximate, although at present, SET COVER has a slightly stronger lower bound. In particular, it is NP-hard to approximate SET COVER to within a factor of c ċ log n for some c > 0. Thus, the MINIMUM RESET SEQUENCE problem is as least as hard to approximate as SET COVER. This improves the previous best lower bound which showed that it was NP-hard to approximate the MRS on binary alphabets to within any constant factor. Our result requires an alphabet of arbitrary size.